The gradient of the policy performance objective, $J(\theta)$, with respect to the policy parameters $\theta$ is derived using the log-derivative trick. This mathematical technique transforms the derivative into an expectation that can be estimated from sampled trajectories. The derivation is as follows: $\frac{\partial J(\theta)}{\partial \theta} = \frac{\partial}{\partial \theta} \sum_{\tau \in \mathcal{D}} \mathrm{Pr}_{\theta}(\tau)R(\tau)$ $= \sum_{\tau \in \mathcal{D}} \frac{\partial \mathrm{Pr}_{\theta}(\tau)}{\partial \theta} R(\tau)$ By multiplying and dividing by $\mathrm{Pr}_{\theta}(\tau)$, we can introduce the gradient of the logarithm: $= \sum_{\tau \in \mathcal{D}} \mathrm{Pr}_{\theta}(\tau) \frac{1}{\mathrm{Pr}_{\theta}(\tau)} \frac{\partial \mathrm{Pr}_{\theta}(\tau)}{\partial \theta} R(\tau)$ $= \sum_{\tau \in \mathcal{D}} \mathrm{Pr}_{\theta}(\tau) \frac{\partial \log \mathrm{Pr}_{\theta}(\tau)}{\partial \theta} R(\tau)$ This final form shows that the policy gradient is the expected value of the score function, $\frac{\partial \log \mathrm{Pr}_{\theta}(\tau)}{\partial \theta}$, weighted by the cumulative reward, $R(\tau)$.